where β is the convective velocity vector, c is the diffusion coefficient, u is a transported scalar, and F is a source term. The underlying finite element discretization method in COMSOL Multiphysics is the Galerkin method. When discretizing Equation 3-3 using the Galerkin method, the resulting numerical problem becomes unstable for an element Péclet number (Pe) larger than one (Ref. 1):

where h is the mesh element size. The Péclet number is a measure of the relative importance of the convective effects compared to the diffusive effects; a large Péclet number indicates that the convective effects dominate over the diffusive effects.

As long as diffusion is present, there is — at least in theory — a mesh resolution beyond which the discretization is stable. This means that the spurious oscillations can be removed by refining the mesh. In practice, this method is seldom feasible because it can require a very dense mesh. Instead, it is common practice to use stabilization methods — that is, methods that add artificial diffusion. The COMSOL products include several such methods, some of which are described in An Example of Stabilization.